[dba-Tech] Three Doors Problem

This is old science, but perhaps it's time to trot it out again, if only
because as I write this, I'm listening to a cbc-radio program called Ideas,
whose current subject is probability. So allow me to trot out a problem
that is a couple of decades old, and invite you to solve it without
referring to Google or Scientific American (in which there occurred a very
heated discussion that lasted about six months in Letters to the Editor,
many of whose authors cited various credentials from prestigious
universities). But in the end, math proved correct.

It's the classic three-doors problem, in the following descriptions of
whose doors, feel free to substitute objects of interest, the only proviso
being that two should be eminently desirable and the third an unpleasant
alternative.

I am the host of a TV show. You are the contestant. You are presented with
three doors.

Lights focus on me, and I announce that our new contestant is the
lovely/handsome Mr./Ms. J.D. Something, a seasoned developer from
Somewhere, Kansas and specializing in several programming languages.

Lights focus on you. In 30 seconds, you describe your current career,
failed marriages, naughty children, and recent auto-accidents. Enough of
the fluff! Time to move on to The Game.

Behind these three doors lie your three prizes, my darling/handsome
contestant!

Behind two of these doors lies something ghastly, Behind the other is the
answer to all your problems, and all your family's problems, forever!

I was going to get into ghastly descriptions of what lies behind the bad
two doors, and a rhapsodic description of what lies behind the third, but
let's strive for simplicity.

1. Choose a door: A, B or C.
2. As host, I select one of the remaining two doors, revealing something
ghastly.
3. I now invite you to switch your choice to the remaining door, or stick
with your original choice.

Does it make a difference whether you stay or switch, and if so why?

Facts:
This question originally arose in the pages of Scientific American about
40+ years ago, IIRC.
It's only fair if you try to answer, and explain your answer, in about 20
lines of text and/or algebra.
I knew the answer even before the SA discussions began, and proved it using
three playing cards, within 5 minutes.
Many of the finest statisticians in the world argued about the answer for
about six months in Letters to the Editor of said publication.

Provisos:
Even though I have cited the source material and its ensuing lengthy
discourse, if you want to play fair, please ignore the sources and deal
with the problem at hand.

Miscellaneous:
I got to the solution rather quickly, but only because I'm an expert at
backgammon, which requires the ability to do a large number of calculations
in a very limited amount of time. Thus I arrived at the correct answer
rather quickly. This is not to say that expertise in backgammon is
required, but that expertise in probability analysis helps.

Proofs are invited. (I haven't written down the solution in algebraic form
lately, but given three cards, I can prove the answer within a couple of
minutes.)

A.